People's decisions depend on the way probability is expressed to them. A lot of financial models assume Normal / Gaussian distribution due to its mathematical tractability. However, as a thin-tailed model, it underestimates the likelihood of extreme events when compared to many real-world phenomena. On the other hand, other distributions, like Student's t or Pareto, can account for extreme values with their fat / heavy tails, which is relevant in the context of modelling the occurrence of outlier events. This property makes it more appropriate for financial returns data (especially at higher frequencies like daily returns), where it's important to estimate the likelihood of very large losses. The difference between the Normal and the other mentioned distributions can be quantified as "unexpectedness", or "surprise.
Standard deviation is a more conservative measure of dispersion for data with outliers, since it tends to give a larger spread. However, MAD can give a better idea of the "typical" deviation, since it's not as influenced by extreme values.
Tesla returns: 1.64% of data is beyond 3 standard deviations
The probability density function (PDF) lines represent the probability density of the returns at each point. The y-axis (right side) represents the density values for these probability distributions. It's important to note that these densities are not direct probabilities but instead give a measure of how much the data is concentrated around a particular value.
A density can take a value greater than 1 because it's not a probability; it's a rate per unit (in this case, per unit return). The height of the PDF at any given point is proportional to the probability density of the returns at that point, assuming the particular distribution.
The fact that some of the values go up to 16 is not an issue. It merely reflects the fact that the data is more concentrated around certain values. The precise values of the density are not often of primary interest; usually, we care more about the overall shape of the distribution.
If you integrate (i.e., find the area under) the PDF across all possible values (from negative infinity to positive infinity), you should get 1. This is because the total probability of all possible outcomes (all possible returns, in this case) should sum up to 1.
In the context of the returns of the Tesla stock, the PDFs provide a way to visualize how the returns are distributed according to different statistical models (normal, t-distribution, etc.). By comparing the shape of these PDFs with the histogram of the actual returns, you can assess the goodness-of-fit of these models.
Let's try to quantify how much a particular price movement deviates from what we would expect under normal circumstances. Assuming TSLA's daily returns follow a fat-tailed distribution, most days will see relatively small price changes, but occasionally, there will be very large changes -- these are the "fat tails".
One way to quantify the "surprise" of a price movement is by calculating how many standard deviations away from the mean it is (or "z-score"). Under a normal distribution, a z-score of 3 or more is considered very rare (occurring less than 0.3% of the time), but under a fat-tailed distribution, such occurrences are more common.
Let's say, for example, the average daily return of TSLA is 0.002% with a standard deviation of 0.036%. If TSLA one day has a return of 0.24%, we can calculate the z-score as follows:
$z = (0.24\% - 0.002\%) / 0.036\% = 6.7$
This suggests the move is 6.7 standard deviations away from the mean, a huge "surprise" under normal distribution assumptions, but in a fat-tailed world, such moves can happen more often than we would normally expect.
Surprise can be measured as the difference between the earnings expectations (Gaussian), and actual earnings (nonlinear, based on a variety of factors including changes in market conditions, the company's strategic decisions, and unforeseen events).
Analysts' expectations are usually reflected in the form of earnings estimates, which are frequently collected and reported by a number of financial information and news services. These include:
Earnings Forecasts: These are typically expressed as an estimate of Earnings Per Share (EPS) for a given quarter or year. The EPS estimate represents the analysts' average expectation of a company's profitability.
Revenue Estimates: Analysts also provide estimates for a company's revenue for a given period. This helps investors understand the expected top-line growth of the company.
Other Financial Metrics: Depending on the company and industry, analysts may also provide estimates for other key financial metrics, such as gross margin, operating margin, EBITDA, and more.
We could look into approximating the concept of "priced in" - the extent to which market expectations, including all publicly available information, are already reflected in the current price of a security. Some of the factors that might be "priced in":
Publicly Available Information: This includes financial reports, news releases, and any other information that's publicly available and pertinent to the value of the asset.
Market Sentiment: This is the overall attitude of investors towards a particular security or financial market. It is the cumulative attitude of all market participants towards risk, and it can heavily influence whether potential news or events are already priced in.
Analyst Estimates: If analysts have widely shared an expectation for a company’s future performance, this expectation may be priced in.
Economic Indicators: Things like GDP reports, unemployment rates, and consumer sentiment indices can all impact what's priced into the market.